Page 47 - Calculus with Complex Numbers
P. 47
where / (t) is a continuous function on the real interval gty /$1. We call t the
parameter, / (t) the parametricfunction gty /$1 the parametric interval. We call
the points a = / ((y), b = / (p) the endpoints of y . We say y is a closed contour if
a = b. The orientation of a contour y is the direction in which the point z = / (t)
moves as t moves along the parametric interval. W e put an arrow on the contour
to indicate the orientation.
Example 1 (Straight Iine) We can parametrise the straight line y going from
a to b as z = (1 - t)a + tb where 0 :jq t :jq 1 (Figtlre 4.3).
Example 2 (Unit circle) We canparametrise the unit circle y described once anti-
clockwise as z = eit where 0 :jq t :jq 2:7r (Figure 4.4).
Example 3 (Unit square) The square y withvertices at 0 1 1+ï i described once
anticlockwise can be written as y = n + yz + p + y4, where n , yz, p , y4 are the
four sides of the square indicated in Figtlre 4.5. W e need a different parametrisation
for each side.
On n W e can take z = t, where 0 :é t :é 1.
On yz W e can take z = 1 + it, where 0 :é t :é 1.
On p W e can take z = t + i, where 1 k: t k: 0.
On y4 W e can take z = it where 1 k: t k: 0.
Ffgure 4.3
Ffgure 4.4
